Proof

That Func(N(C),N(A∘))≃(N(A∘))N(C)Func(N(C), N(A^\circ)) \simeq (N(A^\circ))^{N(C)} is an exponential object in the homotopy category is pretty immediate.

That the left hand is an isomorphic exponential follows from (Lurie, corollary A.3.4.12), which asserts that for CC and DDsSetsSet-enriched categories with CC cofibrant and AA as above, we have that composition with the evaluation map induces a bijection